√788 at a glance
- Exact value
- 2√197
- Decimal (10 places)
- 28.0713376952
- Rounded
- 28.1 · 28.07 · 28.071
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.071338
- Prime factorization
- 2² × 197
- Cube root
- 9.236528
How to simplify √788
Look for the largest perfect square that divides 788. Here it is 4 (2²), because 788 = 4 × 197 and 197 has no square factor left:
The prime factorization tells the same story: 788 = 2² × 197. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 197 stays inside.
Check: (2√197)² = 2² × 197 = 4 × 197 = 788. As a decimal, 2√197 = 2 × 14.0356688476 ≈ 28.0713376952.
Where √788 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √788 lies between 28 and 29. 788 is 4 above 784 and 53 below 841, so the root is closer to 28.
- Straight line between 784 and 841: 28.0702 (0% low)
- Tangent from 28, i.e. 28 + 4 ÷ 56: 28.0714 (0% high)
- Tangent from 29, i.e. 29 − 53 ÷ 58: 28.0862 (0.05% high)
For √788 the tangent at 28 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 788 is just 4 above 784.
Finding √788 with the Babylonian method
Picture a rectangle with an area of 788 and one side x; the other side must be 788 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √788.
Start from the nearest whole number, 28 (28² = 784):
| Step | Guess x | 788 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 28.1428571429 | 28.0714285714 | 4 |
| 2 | 28.0714285714 | 28.0712468193 | 28.0713376954 | 9 |
| 3 | 28.0713376954 | 28.0713376951 | 28.0713376952 | all 10 shown |
The count of correct decimals went 4, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √788 = 28.0713376952 to every decimal shown.
√788 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √788 the pattern is [28; 14, 56] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √788 is irrational.
Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:
| Fraction | Decimal | Error |
|---|---|---|
| 28/1 | 28.0000000000 | 7.1 × 10⁻² |
| 393/14 | 28.0714285714 | 9.1 × 10⁻⁵ |
| 22,036/785 | 28.0713375796 | 1.2 × 10⁻⁷ |
| 308,897/11,004 | 28.0713376954 | 1.5 × 10⁻¹⁰ |
| 17,320,268/617,009 | 28.0713376952 | < 10⁻¹⁰ |
| 242,792,649/8,649,130 | 28.0713376952 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 788y² = 1. Its smallest solution in positive whole numbers is x = 393, y = 14.
√788 in geometry and everyday measurements
- 788 square feet is 73.2 m². Laid out as a square — a small house footprint or a lot — it is about 28.07 ft (28 ft 1 in) on a side.
- 788 = 2² + 28², so by the Pythagorean theorem √788 is the diagonal of a 2 × 28 rectangle — and the distance between the points (0, 0) and (2, 28) on a grid.
- Since √788 = 2√197, a length of √788 is exactly 2 copies of the length √197 laid end to end.
Square roots near √788 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √785 | √785 | 28.0179 | No |
| √786 | √786 | 28.0357 | No |
| √787 | √787 | 28.0535 | No |
| √788 | 2√197 | 28.0713 | No |
| √789 | √789 | 28.0891 | No |
| √790 | √790 | 28.1069 | No |
| √791 | √791 | 28.1247 | No |
- The cube root of 788 is about 9.236528.
- Because 788 = 4 × 197, the root is twice √197: 2 × 14.035669 ≈ 28.071338.
Frequently asked questions
What is the square root of 788?
The square root of 788 is 2√197 in simplest radical form, which is about 28.0713376952. The negative root, −28.071338, also squares to 788.
Is the square root of 788 rational or irrational?
Irrational. 788 is not a perfect square — it falls between 784 and 841 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √788 be simplified?
Yes. The largest perfect square dividing 788 is 4, so √788 = √4 × √197 = 2√197.
What is √788 rounded to two decimal places?
√788 ≈ 28.07 to two decimal places (28.1 to one, 28.071 to three). Check: 28.07² = 787.9249, close to 788.